Finance:Forward rate
The forward rate is the future yield on a bond. It is calculated using the yield curve. For example, the yield on a three-month Treasury bill six months from now is a forward rate.[1]
Forward rate calculation
To extract the forward rate, we need the zero-coupon yield curve.
We are trying to find the future interest rate [math]\displaystyle{ r_{1,2} }[/math] for time period [math]\displaystyle{ (t_1, t_2) }[/math], [math]\displaystyle{ t_1 }[/math] and [math]\displaystyle{ t_2 }[/math] expressed in years, given the rate [math]\displaystyle{ r_1 }[/math] for time period [math]\displaystyle{ (0, t_1) }[/math] and rate [math]\displaystyle{ r_2 }[/math] for time period [math]\displaystyle{ (0, t_2) }[/math]. To do this, we use the property that the proceeds from investing at rate [math]\displaystyle{ r_1 }[/math] for time period [math]\displaystyle{ (0, t_1) }[/math] and then reinvesting those proceeds at rate [math]\displaystyle{ r_{1,2} }[/math] for time period [math]\displaystyle{ (t_1, t_2) }[/math] is equal to the proceeds from investing at rate [math]\displaystyle{ r_2 }[/math] for time period [math]\displaystyle{ (0, t_2) }[/math].
[math]\displaystyle{ r_{1,2} }[/math] depends on the rate calculation mode (simple, yearly compounded or continuously compounded), which yields three different results.
Mathematically it reads as follows:
Simple rate
- [math]\displaystyle{ (1+r_1t_1)(1+ r_{1,2}(t_2-t_1)) = 1+r_2t_2 }[/math]
Solving for [math]\displaystyle{ r_{1,2} }[/math] yields:
Thus [math]\displaystyle{ r_{1,2} = \frac{1}{t_2-t_1}\left(\frac{1+r_2t_2}{1+r_1t_1}-1\right) }[/math]
The discount factor formula for period (0, t) [math]\displaystyle{ \Delta_t }[/math] expressed in years, and rate [math]\displaystyle{ r_t }[/math] for this period being [math]\displaystyle{ DF(0, t)=\frac{1}{(1+r_t \, \Delta_t)} }[/math], the forward rate can be expressed in terms of discount factors: [math]\displaystyle{ r_{1,2} = \frac{1}{t_2-t_1}\left(\frac{DF(0, t_1)}{DF(0, t_2)}-1\right) }[/math]
Yearly compounded rate
- [math]\displaystyle{ (1+r_1)^{t_1}(1+r_{1,2})^{t_2-t_1} = (1+r_2)^{t_2} }[/math]
Solving for [math]\displaystyle{ r_{1,2} }[/math] yields :
- [math]\displaystyle{ r_{1,2} = \left(\frac{(1+r_2)^{t_2}}{(1+r_1)^{t_1}}\right)^{1/(t_2-t_1)} - 1 }[/math]
The discount factor formula for period (0,t) [math]\displaystyle{ \Delta_t }[/math] expressed in years, and rate [math]\displaystyle{ r_t }[/math] for this period being [math]\displaystyle{ DF(0, t)=\frac{1}{(1+r_t)^{\Delta_t}} }[/math], the forward rate can be expressed in terms of discount factors:
- [math]\displaystyle{ r_{1,2}=\left(\frac{DF(0, t_1)}{DF(0, t_2)}\right)^{1/(t_2-t_1)}-1 }[/math]
Continuously compounded rate
- [math]\displaystyle{ e^{r_2 \cdot t_2} = e^{r_1 \cdot t_1} \cdot \ e^{r_{1,2} \cdot \left(t_2 - t_1 \right)} }[/math]
Solving for [math]\displaystyle{ r_{1,2} }[/math] yields:
- STEP 1→ [math]\displaystyle{ e^{r_2 \cdot t_2} = e^{r_1 \cdot t_1 + r_{1,2} \cdot \left(t_2 - t_1 \right)} }[/math]
- STEP 2→ [math]\displaystyle{ \ln \left(e^{r_2 \cdot t_2} \right) = \ln \left(e^{r_1 \cdot t_1 + r_{1,2} \cdot \left(t_2 - t_1 \right)}\right) }[/math]
- STEP 3→ [math]\displaystyle{ r_2 \cdot t_2 = r_1 \cdot t_1 + r_{1,2} \cdot \left(t_2 - t_1 \right) }[/math]
- STEP 4→ [math]\displaystyle{ r_{1,2} \cdot \left(t_2 - t_1 \right) = r_2 \cdot t_2 - r_1 \cdot t_1 }[/math]
- STEP 5→ [math]\displaystyle{ r_{1,2} = \frac{ r_2 \cdot t_2 - r_1 \cdot t_1}{t_2 - t_1} }[/math]
The discount factor formula for period (0,t) [math]\displaystyle{ \Delta_t }[/math] expressed in years, and rate [math]\displaystyle{ r_t }[/math] for this period being [math]\displaystyle{ DF(0, t)=e^{-r_t\,\Delta_t} }[/math], the forward rate can be expressed in terms of discount factors:
- [math]\displaystyle{ r_{1,2} = \frac{\ln \left(DF \left(0, t_1 \right)\right) - \ln \left(DF \left(0, t_2 \right)\right)}{t_2 - t_1} = \frac{- \ln \left( \frac{ DF \left(0, t_2 \right)}{ DF \left(0, t_1 \right)} \right)}{t_2 - t_1} }[/math]
[math]\displaystyle{ r_{1,2} }[/math] is the forward rate between time [math]\displaystyle{ t_1 }[/math] and time [math]\displaystyle{ t_2 }[/math],
[math]\displaystyle{ r_k }[/math] is the zero-coupon yield for the time period [math]\displaystyle{ (0, t_k) }[/math], (k = 1,2).
Related instruments
See also
References
- ↑ Fabozzi, Vamsi.K (2012), The Handbook of Fixed Income Securities (Seventh ed.), New York: kvrv, p. 148, ISBN 978-0-07-144099-8.
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